Numerical evidence of logarithmic regions in channel flow at Re-tau=8000

Direct numerical simulations of channel flow up to Re-tau = 8000 have been performed to determine the existence of a logarithmic region in channel flow at high-Reynolds number. It is notable that the logarithmic variations both in the mean velocity, U+ = (1/kappa)ln(y(+)) + B and the streamwise Reynolds stress, (u+ u+) over bar = B-1 -A(1) ln(y/h) are first confirmed for Re-tau = 8000, with kappa = 0.387, B = 4.21 and A(1) = 1.65, B-1 = 1.23, where y is the distance from the wall and h is the channel half-height. The logarithmic region of U+ locates between 300 < y(+) < 1100 (y/h = 0.14) and that of (u+ u+) over bar locates between 1200 (y/h = 0.15) < y+ < 2000 (y/h = 0.25). Therefore, these logarithmic regions do not overlap with each other. Furthermore, the region of k(x) (-1) law is studied without using the frozen turbulence hypothesis in logarithmic regions of the mean velocity and the streamwise Reynolds stress.